We consider the problem of variable separation for the classical integrable Hamiltonian systems possessing gl(n)-valued Lax matrices satisfying quadratic Poisson brackets of Freidel and Maillet [Phys. Lett. B 262(2-3), 278 (1991)] with spectral-parameter dependent a-b-c-d tensors. We formulate, in terms of the corresponding a-b-c-d tensors, a sufficient condition that guarantees that the separating functions A(u), B(u) of Sklyanin [Commun. Math. Phys. 150(1), 181 (1992)], Scott [e-print arXiv:hep-th 940303 (1994)], and Gekhtman [Commun. Math. Phys. 167, 593 (1995)] produce canonical coordinates. We consider an important subcase of a-b-c-d algebras, namely, the case of classical reflection equation algebras [E. Sklyanin, J. Phys. A: Math. Gen. 21(10), 2357 (1988)], and formulate, in terms of the corresponding r-s-matrices, the analogous sufficient condition that guarantees the canonicity of the constructed coordinates. For the case s = 0, we recover our previous results on the variable separation for quadratic Sklyanin brackets [B. Dubrovin and T. Skrypnyk, J. Math. Phys. 60, 093506 (2019)]. We consider an example of gl(n) ⊗ gl(n)-valued trigonometric a-b-c-d tensors that satisfy the considered condition and find a class of Lax matrices for them for which the obtained set of canonical coordinates is complete.

中文翻译：

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二次代数的变量分离：Maillet 代数和反射方程代数

我们考虑具有满足 Freidel 和 Maillet 二次泊松括号的 gl(n) 值 Lax 矩阵的经典可积哈密顿系统的变量分离问题 [Phys. 莱特。B 262(2-3), 278 (1991)] 与光谱参数相关的 abcd 张量。我们根据相应的 abcd 张量制定了一个充分条件，以保证 Sklyanin 的分离函数 A(u), B(u) [Commun. 数学。物理。150(1), 181 (1992)]、Scott [e-print arXiv:hep-th 940303 (1994)] 和 Gekhtman [Commun. 数学。物理。167, 593 (1995)] 产生规范坐标。我们考虑 abcd 代数的一个重要子案例，即经典反射方程代数的案例 [E. Sklyanin，J. Phys。答：数学。Gen. 21(10), 2357 (1988)]，并根据相应的 rs 矩阵公式化，保证构造坐标规范性的类似充分条件。对于 s = 0 的情况，我们恢复了先前关于二次 Sklyanin 括号 [B. Dubrovin 和 T. Skrypnyk，J. Math。物理。60, 093506 (2019)]。我们考虑一个满足所考虑条件的 gl(n) ⊗ gl(n) 值三角 abcd 张量的例子，并为它们找到一类 Lax 矩阵，其获得的规范坐标集是完整的。